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2021 | Buch

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Upper and Lower Bounds for Stochastic Processes

Decomposition Theorems

verfasst von: Prof. Michel Talagrand

Verlag: Springer International Publishing

Buchreihe : Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book provides an in-depth account of modern methods used to bound the supremum of stochastic processes. Starting from first principles, it takes the reader to the frontier of current research. This second edition has been completely rewritten, offering substantial improvements to the exposition and simplified proofs, as well as new results.
The book starts with a thorough account of the generic chaining, a remarkably simple and powerful method to bound a stochastic process that should belong to every probabilist’s toolkit. The effectiveness of the scheme is demonstrated by the characterization of sample boundedness of Gaussian processes. Much of the book is devoted to exploring the wealth of ideas and results generated by thirty years of efforts to extend this result to more general classes of processes, culminating in the recent solution of several key conjectures.
A large part of this unique book is devoted to the author’s influential work. While many of the results presented are rather advanced, others bear on the very foundations of probability theory. In addition to providing an invaluable reference for researchers, the book should therefore also be of interest to a wide range of readers.

Inhaltsverzeichnis

Frontmatter

Chapter 1. What Is This Book About?

Abstract

We give an overall description of the contents of the book.

Michel Talagrand

The Generic Chaining

Frontmatter

Chapter 2. Gaussian Processes and the Generic Chaining

Abstract

We build the basic tools relating the size of stochastic processes with the geometry of the metric space consisting of the index set provided with a distance which suitably controls the increments of the process. We describe the generic chaining and the construction of suitable sequences of partitions in metric spaces, the characterization of sample boundedness of Gaussian processes, and the size of ellipsoids in Hilbert spaces.

Michel Talagrand

Chapter 3. Trees and Other Measures of Size

Abstract

We describe the size of a metric space through the size of the trees it contains, for various notions of trees. We introduce Fernique’s functional, an equivalent measure of size, which will be basic for proving results in the non-Gaussian case.

Michel Talagrand

Chapter 4. Matching Theorems

Abstract

Based on the structure of ellipsoids in Hilbert space, we prove the classical matching theorems between independent random samples of the unit square, the Ajtai-Komlos-Tusnady matching theorem and the Leighton-Shor grid matching theorem, and we explain the occurrence of the unusual powers of logarithms there.

Michel Talagrand

Some Dreams Come True

Frontmatter

Chapter 5. Warming Up with p-Stable Processes

Abstract

We outline our general scheme of proof of lower bounds in the simple case of p-stable processes by combining the fact that they are conditionally Gaussian with our results on Gaussian processes.

Michel Talagrand

Chapter 6. Bernoulli Processes

Abstract

We describe the basic tools bearing on Bernoulli processes, which parallel those of the Gaussian case. A Bernoulli process is a family of random variables (r.v.s), each of which is a different linear combination of the same independent random signs. There are two obvious ways Bernoulli process may be bounded: First, it may happen that the corresponding Gaussian process (where the random signs are replaced by standard independent Gaussian r.v.s) is already bounded. Second, it may happen that the sum of the absolute values of the coefficients of the random signs is already bounded. We state the Bernoulli conjecture, which asserts that every situation is a combination of these.

Michel Talagrand

Chapter 7. Random Fourier Series and Trigonometric Sums

We solve the long-standing problem of finding necessary and sufficient conditions for the convergence of random trigonometric series with independent coefficients. Metric spaces do not suffice for this purpose; one has to consider sets endowed with sequences of distances. The necessary and sufficient conditions take the form of a smallness condition of the underlying compact group for such a family of distances. We prove our first decomposition theorems. In words, one of these theorems states that if a random Fourier series converges a.s., it can be decomposed in the sum of three random Fourier series, each of which converges a.s.

Michel Talagrand

Chapter 8. Partitioning Scheme and Families of Distances

Abstract

We extend the method constructing suitable increasing sequences of partitions from the metric spaces of Chap. 1 to the setting of families of distances. This makes it possible to extend the necessary and sufficient conditions for sample boundedness of Gaussian processes to vastly more general classes of processes.

Michel Talagrand

Chapter 9. Peaky Part of Functions

Abstract

Given a set of functions which, in a precise sense, is not too large, we learn how to decompose in a coherent way each function of the set in the sum of its peaky part and its tamed part. These results are technical but are the key to much of what follows.

Michel Talagrand

Chapter 10. Proof of the Bernoulli Conjecture

Abstract

We prove the Bernoulli conjecture.

Michel Talagrand

Chapter 11. Random Series of Functions

Abstract

In this chapter we consider the boundedness of random series of functions where the summands are independent symmetric. There are two natural ways such a series may be bounded. First it may happen that the sum of absolute values of the summands is already bounded. Or it may happen that the boundedness is witnessed by a natural chaining argument. We prove the decomposition theory stating that every situation is a mixture of these cases, solving several 30 years old conjectures.

Michel Talagrand

Chapter 12. Infinitely Divisible Processes

Abstract

We study infinitely divisible processes, both in the harmonic and the general setting. The structure of these processes resembles that of certain random series of functions. We obtain necessary and sufficient conditions for boundedness in the harmonic case. For the general case, we prove a decomposition theorem in the same spirit as the decomposition theorem of the previous chapter.

Michel Talagrand

Chapter 13. Unfulfilled Dreams

Abstract

We state a number of daring conjectures concerning the boundedness of random series of functions with non-negative summands are the special setting of selector processes.

Michel Talagrand

Practicing

Frontmatter

Chapter 14. Empirical Processes, II

Abstract

We provide a sample of the deep methods used to bound empirical processes by proving two recent results.

Michel Talagrand

Chapter 15. Gaussian Chaos

Abstract

We study order 2 Gaussian chaos and the tails of higher-order chaos.

Michel Talagrand

Chapter 16. Convergence of Orthogonal Series: Majorizing Measures

Abstract

We study order 2 Gaussian chaos and the tails of higher-order chaos.

Michel Talagrand

Chapter 17. Shor’s Matching Theorem

Abstract

We prove Shor’s matching’s theorem, a deep refinement of the Ajtai-Komlos-Tusnady theorem, and we describe the ultimate matching conjecture which would unify all the classical matching results in two dimensions.

Michel Talagrand

Chapter 18. The Ultimate Matching Theorem in Dimension 3

Abstract

We prove a version of the ultimate matching conjecture for matchings in three dimensions and more.

Michel Talagrand

Chapter 19. Applications to Banach Space Theory

Abstract

We apply some of the previous ideas to the theory of Banach spaces. We study the cotype of operators from a space of continuous functions to a general Banach space. We perform several probabilistic constructions in Banach space theory, including Bourgain’s celebrated solution of the Lambda-p problem. We present new results of Pisier on Sidon sets.

Michel Talagrand

Backmatter

Titel: Upper and Lower Bounds for Stochastic Processes
verfasst von: Prof. Michel Talagrand
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-82595-9
Print ISBN: 978-3-030-82594-2
DOI: https://doi.org/10.1007/978-3-030-82595-9